Multipole Charge Conservation and Implications on Electromagnetic Radiation

Home , Gauge fixing, Lorenz gauge condition

JHEP06(2017)080 Springer June 7, 2017 June 15, 2017 : : November 7, 2016 February 18, 2017 : : Accepted Published Revised Received Published for SISSA by https://doi.org/10.1007/JHEP06(2017)080 . 3 1610.02870 The Authors. [email protected] c Gauge Symmetry, Global Symmetries, Space-Time Symmetries

It is shown that conserved charges associated with a specific subclass of gauge , ali School of Physics, InstituteP.O.Box for 19395-5531, Research in Tehran, Iran Fundamental Sciences (IPM), E-mail: Open Access Article funded by SCOAP Keywords: ArXiv ePrint: there is a novel equipartition relationwhile between the conserved, two can contributions. freely Theand multipole interpolate therefore charge, between can the be source propagatedtion, with we and obtain the the infinite radiation. electromagnetic numberof Using of field, constraints the charged over matter. multipole the charge radiation conserva- produced by the dynamics symmetries of Maxwell electrodynamicstipole are moments. proportional The to symmetriesgauge, the are with well residual nontrivial known gauge conserved electric transformationsceive charge surviving mul- at contributions the both spatial Lorenz infinity. fromis the These nothing charged “Multipole but matter charges” the and re- electric electromagnetic multipole fields. moment The of former the source. In a stationary configuration, Abstract: electromagnetic radiation Ali Seraj Multipole charge conservation and implications on JHEP06(2017)080 ]. This can also 1 ]) which states that the 1 6 12 7 19 3 3 – 1 – 16 12 10 14 8 5 13 14 11 14 1 17 4.1.1 Screening4.1.2 effect and equi-partition relation Symplectic symmetries Dividing symmetries of a theory into global or local, the above theorem is usually 5.3 Infinite constraints over radiation 4.2 Magnetostatics 5.1 Conservation 5.2 A preliminary example 4.1 Electrostatics 2.1 Charges in2.2 the covariant phase Noether space current 2.3 Residual gauge symmetries and asymptotic symmetries 3.1 The nontrivial sector supposed to be restricted toconsidered the in former. the Local Noether’s (gauge)existence symmetries, second of on local theorem the symmetries, (also other impliesBianchi discussed hand, a identities. are in set [ of constraints for the theory, usually known as In her seminal paper, E.constants Noether of motion established in a the profoundbe action link rephrased formulation between of in the particle the symmetries orfunction and Hamiltonian field over description theories the where [ phase aalso space symmetry generates which can the be commutes symmetrycontinue with associated transformation to the with hold through a Hamiltonian in the of the Poisson quantum the bracket. theory system if and These the results symmetry is anomaly free. 1 Introduction 6 Discussion A Algebra of charges in covariant phase space 5 Electrodynamics 4 Stationary configurations 3 Residual gauge symmetries of Maxwell theory Contents 1 Introduction 2 Gauge symmetries and conservation laws JHEP06(2017)080 ]. 5 8 we – 5 4 ]. The 4 – 2 residual gauge we derive in a systematic way, 2 we determine the residual symmetries as 3 – 2 – . An appendix is devoted to the Poisson bracket of 6 In section 1 ). 2.20 ]. ) and ( 15 , 3 respecting a given gauge condition. Such viewpoint was stressed in [ 2.18 The organization of the paper is as follows. In section Multipole moments in electrodynamics are obviously not conserved. For example a On the other hand, one may put the boundary conditions on gauge invariant quantities However, simultaneous implementation of both of the above Noether theorems leads Although in Maxwell theory, these results can also be obtained by the usual Noether’s approach, but 1 the radiation. We conclude incharges section over the covariant phase space. in general the Hamiltinianof approaches the are correct preferred charge since [ e.g. in gravity the Noether charge is only a part phase space method.easily Those jump who to are ( the not physical subset interested of in U(1)compute gauge the transformations -in details of an Maxwell of electrostatic theory.and the configuration- Then show derivation in our the section main can charges result associatedwe relating the discuss with charges electrodynamics these and and electric symmetries show multipole moments. how the In above section conservation laws put constraints on that this charge is nothing butelectrodynamics. the The conserved charge multipole associated charge withand can residual the freely symmetries electromagnetic interpolate of field. between the charged matter the conserved charges associated with nontrivial gauge symmetries, using the covariant point charge located atleave the origin origin, of it space will“soft obtain has multipole dipole charge” only is and attributed monopole higher tocomposed moments. the moment, of electromagnetic However, while field, we hard the will if and total show it multipole soft that charge starts pieces if a will to be a conserved quantity. Interestingly, it turns out transformations In this paper,transformations we can will be follow this associatedMaxwell electrodynamics, approach with and we nontrivial will show conservedinterpretation in show that charges. terms that a of In such subclass electric the conserved multipole of moments. quantities context residual have of gauge a very nice like the field strength,observable instead quantities of are gauge gauge invariant. fieldon Such the itself. boundary allowed This gauge conditioncan impose can transformations. be no be However, singled restriction more still out a physical,the by subclass since gauge choosing of fixing the a condition gauge gauge kills transformations condition most instead of the of gauge a redundancies, boundary it condition. allows for While of physical interest.surface A integral charges more nonvanishing. relaxed However, onedoes boundary should not condition make sure lead on thatrelaxed such to gauge relaxation boundary fields, divergent conditions charges. having can nonvanishingasymptotic make charges, form The symmetry the a algebra. “large closed algebra gauge called transformations” the allowed by the to the association ofkey a result conserved in charge2) to this integrals, a case, instead local is of (gauge)fast that volume enough symmetry near (codimension the as the 1) well boundary, chargesWhile the integrals. [ can this charges associated is Accordingly, be usually with if presumed local formulated in symmetries the quantum as would fields field vanish. surface theory, drop it (codimension is not the case in many examples JHEP06(2017)080 (2.5) (2.4) (2.1) (2.2) (2.3) must be µ j ). Taking another φ 1 form of spacetime − d . ) ] which is well established. (as a ψ , 1 12 ω δ – ( = 1 and the Largrangian takes 9 c Θ matter 2 δ L = stands collectively for all dynamical − 0 + , ) µ . µ ) and the matter field ψ ψ . ν A µ 2 = j δψ µ δ ( A j ( 0 = = 0 ε ] for a review). This is the setup we use in Θ Θ − d µ 1 µν 17 – 3 – j δ µν µ ≈ F ∂ µ F is the field strength, and the current L ∂ ) = δ µν µ ψ F 2 A 1 4 ν ∂ − ψ, δ 1 ] (see also [ − = leads to the Maxwell field equations defines the symplectic current ν 16 L µ ψ, δ – A Θ ( A µ ω ∂ 13 , ≡ 3 , ] 2 ν A µ [ ∂ = is the Lagrangian as a top form, and µν L F According to the action principle, the on-shell variation of the Lagrangian is by con- To start, one needs to define a symplectic form on the space of field configurations. fields in the theory (inantisymmetric our variation case of the gauge field and a two form over the phase space) struction a total derivative where The symplectic form can(of dynamical then fields). be Moreover, one usedwhich can generates to that associate gauge a define transformation Hamiltonian through aof to the the each Poisson Poisson bracket. Hamiltonian gauge bracket symmetry, The will between on-shell define value functionals the charge of that gauge symmetry. However, this has the drawbackpotentially that leads it to breaks cumbersome theconstruction expressions. called manifest Instead, the covariance one “covariant of phase can theconserved space” use theory, charges to a and study [ pretty gauge mathematical symmetriesthis and paper. associated gauge symmetries. 2.1 Charges in theIn covariant order phase to space becan able to use study the the Hamiltonian conservation laws formulation associated of with gauge gauge theories symmetries, one [ Variation with respect to In the next section, we give a systematic approach to compute the charges associated with where conserved We consider the theory of Maxwellfield. electrodynamics sourced by We an choose arbitrary the chargedthe matter natural form units in which 2 Gauge symmetries and conservation laws JHEP06(2017)080 ψ λ δ (2.6) (2.7) (2.8) (2.9) (2.10) (2.13) (2.11) (2.12) + ω ψ → (either local ψ ψ λ δ . As we will see, + c.c ψ , ) + → ) φ x µ ψ , ( D φ , . ( ) ) ∗ ) , x c.c ψ µ ( , ∗ 2 ) ) + dx i e φ φ ieλ µ . . µ ψ, δ = δφ ω ) ) 1 ( D ∗ ψ ( µ ) ψ, δψ ? ) = ( λ j φ φ x λ ψ, δ µ µ = ( ( ψ, δψ k φ ( D D ω ω d λ ( λ Σ + k Z ψ, δψ, δ + Σ ) = µν – 4 – ∂ ν , ψ I F ) λ ) = , δ µ ) δA µν = Ω( ψ = ) is defined through the symplectic current x 2 F ( λ dx ψ λ µν 2 1 4 µ λ F θ µ ψ, δ δH − δH ( ∂ 1 ψ, δψ, δ ? ψ, δ ( = 1 ) = ω = ψ, δ ) = L δψ x ψ, δ Θ ( ( Ω( µ µ θ A and its current is given by λ δ µ ieA ] that for a gauge transformation in a gauge invariant theory one has + 3 µ , ∂ 2 = ) is a local function (or tensor) that parametrizes the gauge transformation. x µ ( D λ The theory of scalar QED is invariant under the transformations Now let us compute the charges corresponding to gauge symmetries of Maxwell elec- The Hamiltonian associated to a symmetry transformation A gauge theory involves local symmetry transformations of the form It can be checked that paper is general for any type of electrically charged matter field. To compute the charges, let us define the dual quantities trodynamics. For explicit computation, let us take the Lagrangian of scalar QED where the result is independent of the specific form of the matter field, and hence the rest of the the Gauss’ surface integral. Onare the given other by hand volume it integralswhile shows in in why Special General energy Relativity, Relativity and where (where diffeomorphisms angularare Lorentz are momenta given local symmetries by symmetries, are surface similar global) integrals. quantities Therefore one finds the importantsymmetry result is that given the by conserved a charge co-dimensionsymmetry associated 2 integral. with is a given Meanwhile gauge the by conserved a charge volume of a integral. global This explains why the electric charge is given by that is the symplecticexact form. current Accordingly contracted with a gauge transformation is necessarily an or global) is then defined through It is proved [ over a spacelike hypersurface Σ in spacetime. where The pre-symplectic form Ω( JHEP06(2017)080 ) 2.10 (2.22) (2.19) (2.20) (2.21) (2.14) (2.15) (2.16) (2.17) (2.18) ) provided ) to arrive at . 2.18 2) c.c. 2.15 . + ↔ ) x ( (1 λ δφ ν ∗ − , ) ∂ ) φ µν µ ). Using gauge transforma- c.c F D ψ µ + λ , Σ ∗ on-shell ) + ( . d φ x ) 2 φ ( Σ λ , δ ∗ x , λ Z ) ) ( ν µν φ φ λ , ) ψ, δψ, δ ∂ = µ ) µ ( x µν ) ( µ δF . µν D s D µν ( λ λ ( ω ( F . F 1 µν δ µ δ µν , where = 0 µν Σ + ) λδF F s d Σ µ ( µ λ ( λ + ( ieλ λδj d ν j – 5 – Σ J ν Q ) ∂ ν ∂ µ ∂ Σ ,Q + − x ). Accordingly the charges are defined I ∂ ∂ µ A ( + 2 j I λ λ = λ ) ) = δ ν ν ) = 2.8 h µ λ λ ( x ∂ ∂ = λ ψ ( µν J λ Q λ λ µν µν ≡ − F ). µ δQ 1 Q = µ λ δ Σ δF δF J λ 2.1 d = Q Σ ) = ψ, δψ, δ ) = Z on-shell ( ψ ψ µ 2 − λ ω = ) and the corresponding charge can be decomposed into hard and soft ψ, δ ) 1 h ( λ Q 2.19 ψ, δψ, δ ψ, δ ( ( . µ µ 5 ω ω ), we arrive at 2.11 The current ( pieces, which include the contributionfield of electrically respectively. charged Explicitly fields and the electromagnetic This continuity equation ensuresthat the there is conservation no ofthis flux charges in of section defined current at in the ( boundary of spacetime. We will elaborate more on which is a total derivative and accordingly conserved 2.2 Noether current One can simply integratethe over Noether the current variation in the symplectic current ( We stress again that thehave charges explicit are dependence on written matter only fieldsand in and work terms from in of now the on the general we gauge forget case field the ( and Lagrangian does ( not which confirms the general theorem ( and integrating over variations, one finds the finite charges Using the Maxwellsimple equations form for for the the symplectic linearized current perturbations, we obtain the following To compute the charges,tions we ( need to compute and hence JHEP06(2017)080 , ). 5 µν F (2.23) (2.24) ]. Also 30 , 29 are the spatial B , . It turns out that , surface, we can use ) E ] and more generally x ( 25 λ ]. In QED and gravity, – const 21 · ∇ = 22 . Note that throughout this , where , t E k 7 x 3 B , d → ∞ 2 ijk Ω Z R d − 2 − r = = + ) ij 2 s F ( λ dr residual gauge symmetries + . An asymptotic symmetry is defined through a and – 6 – ], microscopic counting of black hole entropy [ 2 i ) and not on gauge invariant quantities (like dt ] and its gravitational counterpart [ µ E 18 ρ, Q , − A ) λ , − 28 x – 10 = ( = E 2 26 · i xλ 0 3 ds d~a F d S ), I Z j − − ρ, = = asymptotic symmetry ) = ( λ h ( λ µ Q j ]. The role of such symmetries in theories with Weyl scaling is still Q 32 , 31 ]. without explicit latin index refers to the three dimensional spatial gradient. can be chosen as a sphere of constant raduis 33 S ∇ ], and even an identification of black hole microstates [ We should make a comparison here between the notions of residual gauge symmetry 20 , have been obtained from thismust approach, be it imposed has on the gaugeHowever, drawback fields that (like the the physical boundary significance conditions physics involves of gauge such invariant boundary quantities.that case conditions The such is situation conditions not can is be clear, different interpreted in since as gravity, the local since choice in of “observers” at infinity. and the more familiar consistent boundary condition on thegauge gauge transformations fields. which Boundary conditions break(trivial) rule the if out the BCs. too associated large defined charge The as is remaining the nonvanishing ones, (vanishing). quotientLie of are Asymptotic bracket, nontrivial symmetries called they modulo are nontrivial form trivial the gauge asymptotic transformations. symmetry algebra. Through the While many intriguing results Low’s subleading soft theoremslarge [ gauge transformations areHall used effect to [ describeunclear the [ so called “edge states” in quantum different ways that “large gaugetheories. symmetries” can Most play important famously,basic physical in understanding role the of in holography context different [ 19 of gravity, large gaugethey transformations are provide recently used to prove Weinberg’s soft theorems [ Existence of gauge symmetriestheory, in at a the cost gaugeis of theory bringing then provides in a removed an covariant throughmay infinite description “gauge survive redundancy of this fixing”. in the gauge thea fixing However, system. subset which a of we This specific call residual redundancy class symmetries, of can be gauge “large” symmetries near the boundary. It is argued in many where paper 2.3 Residual gauge symmetries and asymptotic symmetries electric and magnetic fieldsbe respectively. simplified Accordingly, to the expressions for the charges can though we expect that thewithout arguments much can effort. be If generalized wethe to take identifications asymptotically the flat hypersurface spacetimes Σ to be the In this paper, we consider the four dimensional flat spacetime with the metric JHEP06(2017)080 ) = 0 (3.1) (3.2) (3.3) (3.4) (3.5) x ( f 2 2 ∂ ∂x The general 2 , ) ], proving that the θ, ϕ 6= 0 and those with ( . However, there are 34 µ ω ∗ j `,m Y = µ ) A kr ( ) − ( ` , h ) . x ( `,m ω . . B = 0 λ ω iωt λ = 0 ) + ) − ) = 0 µ 2 e x kr ω – 7 – A ( ( µ λ + ) = (+) ` ∇ 2 x h in Fourier modes ∇ t, ( ( λ `,m ) = 0 has oscillatory solutions while the solutions to λ x A ( f ) ` 2 − ω ` = X + m 2 2 ∂ =0 ∂x ∞ ` X ) = x ( ω . λ x ]). An interesting relevant result appeared recently in [ 6 To remove the infinite redundancy in Maxwell theory due to U(1) gauge degrees of On the other hand, if the boundary conditions are imposed on gauge invariant quanti- Similarly the equation ( 2 = 0. They have drastically different behavior at large radial coordinates. grow linearly in There are two qualitativelyω different sets of solutions:solution to those the with wave equation with nonvanishing frequency is Expanding the time dependence of the equation becomes the Helmholtz equation In this gauge, Maxwellstill equations residual become gauge wave transformations equations killed that by respect the the gauge Lorentz fixing. gauge They and satisfy therefore the equation are not In this section, wegauge. determine the Then we residual single gauge out symmetries the of nontrivial Maxwell sector theory offreedom, in these we Lorenz impose symmetries. the Lorenz gauge condition found in [ independence of classical physicsAnti-BRST from symmetry the of the choice corresponding of quantum gauge theory. condition is3 equivalent to the Residual gauge symmetries of Maxwell theory other. This hasthis interesting approach implications can that havethe we its choice will own of gauge drawbacks. discuss conditionsome in which The gauge cannot our form conditions be problem. of are singledfield. out residual favored However, by e.g. symmetries We physical by expect depend considerations. requiring that on Still however causality the a in results general the must propagation proof be of of eventually gauge such independent claim of remains the an gauge open condition, problem (more comments can be ties, no gauge transformation is ruledgauge out symmetries by the are boundary the conditions.gauge nontrivial Then condition. residual the “physical” This symmetry hasmined transformations the all-over that the advantage respect bulk, that and the thethe not form only radial of at dependence the residual boundary. of symmetries Unlike these are the deter- symmetries asymptotic symmetries, can be completely different from one to an- JHEP06(2017)080 ) µν F 3.5 (3.9) (3.6) (3.7) (3.8) (3.10) . To compute this . To this end, we note 0 r . , → ∞ F ) is trivial in the sense that ∗ ) = 0 whose solutions are R `,m . = x Y 3.5 − `,m ( λ λ E 2 2 +1) · 1 It is important to note that this x in order to cancel the minus signs ` , ( − `,m r ∇ 3 c − ) whose solutions are given by ( r `,m `,m O + λ − λ . 3.2 + ) = + `,m E 2 1 ix r · λ ( − `,m e d~a + `,m c x +1 transformations are allowed. This makes our ∼ O I ` – 8 – ` ) i − − E ` , λ `,m ∓ · = X ( = λ ˆ r m ∗ `,m Y =0 `,m ∞ ` ) = ` X r Q x ( − − ) ( ` = h ) = x ( + `,m λ λ ). Therefore assuming that the sources are localized, the reasonable solves the same equation as ( /r are respectively the spherical Hankel functions of first and second kind, (1 E ). We will see shortly that while the negative power modes are again triv- ) · O − r ( . 2.24 /r , h ), we can associate a charge to each of the above residual symmetries (+) h 2.24 Note that in electromagnetic radiation, only the transverse components of the electric and magnetic 3 boundary condition does not imposeis any gauge restriction over invariant. gaugeapproach transformations, Therefore different since with all the of usual asymptotic symmetry group analysis. field fall off as 1 This can be consideredcally interesting as situations a including Neumann radiating systems. boundary condition which allows most of physi- where the integral isintegral, taken we need over to a specifythat sphere the the of asymptotic scalar constant behaviorwhich of raduis falloff ˆ like boundary condition is 3.1 The nontrivial sector Using ( Note that a minus sign isappearing conventionally in absorbed ( in ial, the positive modes have non-vanishing charge with interesting physical interpretation. where As we will see in thethe next section, corresponding any charge solution of is thefrequency vanishing. form of modes, Meanwhile, ( the which storygiven satisfy is by different the for Laplace the equation vanishing which express outgoing and ingoing waves,functions respectively. are The given asymptotic by expansion of these in which JHEP06(2017)080 ) ), and 3.10 2.11 (3.13) (3.14) (3.15) (3.11) (3.12) − `,m λ is given by (for λ . ) ψ 2 ) is vanishing. That is, λ . . 4 3.7 ∗ . ]. ψ, δ E `,m . 1 · 0 Y , as the minus sector is trivial. ), and the transformation rules ( λ 35 ` in ( r d~a → + `,m = 0 ψ, δ E ) 2.14 Q } ( − `,m I · . We will come back to this point later in r 1 2 λ ω ( λ π A = 0 without affecting the physical residual symmetries. d~a 4 Σ 1 0 – 9 – Z ,H = A √ 1 I ∼ O λ = ], the same conclusion was arrived at, using another ω = ≡ dλ H λ } 7 0 { 2 , Q + λ + 0 `,m ]) Q A Q ,H 17 1 λ → H transformations grow badly in large radius and one may expect of symmetry transformation parametrized by { A λ corresponds to We show in the appendix that the Poisson bracket between the + `,m H π λ 1 4 = `,m + 0 , λ + 0 ) to arrive at λ 3.9 . ) in ( 4.1.2 3.7 Algebra of charges. Therefore the conservation of electric charge is a direct consequence of gauge invariance On the other hand, As was promised, we can now show that only the “soft part”, i.e. the zero frequency are pure gauge transformations, since their charge cannot be used to label different Also it is interesting to note that, in vacuum, the above pure gauge transformations can be removed fur- 4 ω we find that the Poisson bracket of charges vanish. ther by the additional gauge fixingBut conditions this like is not what we follow here. a more detailed discussion, see [ Using the symplectic form if Maxwell theory ( Note that hereafter, we dropWe close the this plus section index with of two comments: Hamiltonian generators of the theory. However,gauge this invariance. is not Thiscorresponding the is to whole what information we that show can in be the inferred next from sections the by computing the charge This implies that totalelectromagnetic electric theory. charge is Notethe the that gauge charge field of this intact, constantsection gauge gauge i.e. transformation transformation is of special, since it leaves harmonics kills all divergent terms andThe leads zero to mode well defined, physically meaningful charges. λ configurations of the phase space.argument In based [ on the notion of adiabatic modes [ that the corresponding charges diverge. However, as we will see, the existence of spherical subset of the residualand ( symmetries lead to nontrivial charges. It is enough to use ( Similar reasoning implies that the charge of JHEP06(2017)080 ) 2 i ] and B 0 (3.18) (3.19) (3.17) (3.16) 36 − , λ 2 i E 16 , ( 2 x 3 d vanishes at the R λ . It can then be 1 2 to all local gauge R = . L = 0 λ 2 . orthogonal ∇ ) 2 0 , , λ ) x λ 1 3 A λ i d ( C Z A, ∂ + ( − , 0 ] 2 V 0 λ i = 0. Therefore, the orthogonality condition ,λ − 1 = 0 λ λ R λ ) One may wonder if the physical symmetries [

λ ∂ λ ˙ i 0 A 2 – 10 – H · ∇ λ ˙ x∂ ∇ A, = 3 ( da d } g R 2 . That is, the Lagrangian induces a positive definite 1 2 i = 0, in which the Lagrangian λ Z I A 0 = 2 = ,H A δ 1 ) = L i ], we will perform a general analysis of the space of vacua of λ A A 0 H 37 1 λ { x δ 3 A, δ d λ ). However, analysis of adiabatic motion on the space of vacuum δ R x ( ] for details. ( g λ 37 ) = A 2 A, δ 1 on the space of field configurations. δ ( g g The residual gauge symmetries in the static gauge are parametrized by time inde- Let us fix the static gauge Physical symmetries in static gauge. 4 Stationary configurations In this section, we computea the stationary charges corresponding solution to of nontrivialthese Maxwell residual symmetries. theory. symmetries in This will provide us a physical interpretation of implies whose non-singular solutions coincide with the physical symmetries of Lorenz gauge. The first term in the last line vanishes as configurations reaveals that those residualboundary symmetries are whose unphysical parameter (pure)take gauge the transformations. (IR regulated)argued Let boundary that us physical to denote gauge betransformations symmetries these given with are by respect by those to a which the sphere are metric of introduced above. radius Explicitly where metric pendent functions that the same symmetries appearoriginal in reference the [ static gauge, and refer the interested reader toobtains the the form the level of charges. obtained above crucially dependgauge conditions on as the well. choice InYang-Mills [ of theories Lorenz in the gauge static or gauge. they Here, appear we in will other briefly report some results showing This agrees withextension the of general the Lie theorem algebra that of symmetries the up Poisson to bracket possibly a of central charges extension [ isHere a the Lie central algebra of U(1) gauge symmetries is trivial and no central extension arise at The charges, being the on-shell value of the Hamiltonian generators, obey the same algebra. JHEP06(2017)080 ). . , we 0 (4.7) (4.4) (4.5) (4.6) (4.1) (4.2) (4.3) `, m m,m δ 0 ), . `,` ) δ we have 2.22 θ, ϕ R ) = ( multipole charge . ∗ `,m θ, ϕ Y ( `,m the ) q ∗ `,m Y `,m θ, ϕ . . ) = ) ( 0 Q ) as defined in ( ,m `,m 0 θ, ϕ θ, ϕ ` q . ( ( θ, ϕ 0 Y `,m . ( ∗ Ω ∗ is the vacuum permittivity. Q ,m `,m `,m 0 + 1 d ∗ ` . `,m ` 0 Y `,m Y ` ε ` ` Y Y Y 2 r I ` Ω `,m r q d This gives the classical interpretation of − Φ π ) + 1 +2 x ρ r 4 ` H x 3 ` 5 where = ( . · ∇ d 2 R + 1 Φ. The potential obeys the Laplace equation ) ) an electrostatic configuration of the above + 1 0 h ` ε x ρ `,m Z ` ( `,m d~a +1 q 2 3 `,m ` + 1 π – 11 – Q d q −∇ r 0 4 = 3.13 I +1 ` = +1 ` ` − 2 = 2 Z ∗ − `,m 0 `,m `,m X Y = `,m E = +2 = ` `,m ,m 0 π 0 1 ` Q ` 4 R Q `,m q R `,m x ρ λ `,m Q Φ q 3 = r Q d ) = ∂ ) s x + 1) ( `,m Z 0 Ω ` d Q Φ( − ( 0 2 = R ,m 0 X ` ) is also important. For a better understanding of this factor, let’s h are called the “electric multipole moments” which are determined by I ( `,m π electric multipole moments 1 4 − Q +1 +1 `,m ` ` 2 = = q = 0 and we can write `,m B Q ) and the fact that the integral is taken over a sphere at large 4.1 The factor In SI units, the above would read 5 Therefore the hard piece of chargeHowever, exactly there reproduces is the multipole also moment a of order contribution ( from the fields, i.e. the soft piece compute the hard and soft contributions to the charge This result implies that theproportional charges associated to with the physical residualresidual gauge symmetries symmetries are of electrodynamics. Accordingly, we call Given the orthogonality of sphericalconclude harmonics our main result Using ( Hence an electrostaticments. configuration is Note completely also thattances. determined, higher given Now order the let’s momentsform, compute multipole fall the off mo- charges more ( and more rapidly in large dis- The coefficients the distribution of charged matter as In this case, whose solution can be expanded as 4.1 Electrostatics JHEP06(2017)080 (4.9) (4.8) is the ) that ]. ) it can 12 4.7 40 2.9 , 39 ), ( screening effect 2.8 ], i.e. the smallest sphere 42 [ compared to the bare (hard) , . It is tempting to find a deeper `,m 1 2 = 0 Q , − µ λ ) J horizon h ( `,m ), is not vanishing locally, its volume µ Q Σ ) the current vanishes identically out- d 2.20 the source. The reason is that, while the . Therefore, symplectic symmetries appear corresponding to the total electric charge 12 + 1 ` was defined by the condition that the sym- 0 2.19 , Σ ` 0 0 , Z 2 0 Q – 12 – − = Q 1 = S

) carry higher multipole charges. 1 this approaches s Q ( `,m containing − Q 2 ` does S

Q ]. ) means that in the equilibrium, there is an 41 4.7 symplectic symmetry ) or its finite version ( . The above result can also be written in the suggestive form ) 2.16 ) vanishes locally outside sources. Accordingly, using ( h ( `,m Q 2.16 = 0, which corresponds to the total electric charge, we see from ( ` are two spheres with different radii, both containing the source, and Σ 2 ] the notion of , 1 40 S – , while the soft piece is the contribution from the surrounding electromagnetic field. However, this does not continue to hold when radiation enters in the game, which For other multipole charges, the situation is different. In the electrostatic case, it can Among the multipole charges, only The minus sign in ( 38 `,m only gets nontrivial contributioncontaining within sources. the charge carries nontrivial multipole charges except in the nondynamical sector of the phase space in accordance with results of [ where volume enclosed between the two spheres. Therefore the soft piece of multipole charge be checked that althoughbe they computed are at not anysymplectic symplectic closed current in sphere ( theintegral strict over sense, regions free but of they charged can matter vanishes, still i.e. be easily shown that thenot charges only can at be the computed boundary at [ any surface sorrounding theis sources, precisely symplectic,side since sources. according to ( 4.1.2 Symplectic symmetries In [ plectic current ( which resembles a special “equipartition”multipole relation charge. between Note the that softunderstanding for and of hard this pieces of equipartitionelectromagnetic the relation field and of matter multipole source. charges in equilibrium between from the fields reducingmultipole the charge effective multipole charge As we mentioned,Q the hard pieceFor is the the case contributionelectromagnectic from field matter does fields notU(1) to carry gauge the any theory. electric charges However, it charge. This is what we expect from a 4.1.1 Screening effect and equi-partition relation JHEP06(2017)080 ] ) 43 is a . λ?F F (4.12) (4.13) (4.15) (4.11) (4.14) (4.10) 5 ( d conserved = ] λ 46 J off-shell where [ , ) is any combination of M ) x ( Φ x . . ( λ ) ) λ j −∇ B × θ, ϕ = 1 vanishes. In the following, · = ( r ( λ = 0. The charges can accordingly ), but without the hard contribu- B da `,m S Y ∇ · F. dF I Such charges were introduced in [ 2.24 λ . `,m ∧ 6 ∗ π + 1 `,m M 4 · ∇ ) = Y ` ) detect the electric and magnetic distri- dλ ` x 2 ( B . In differential forms language where λ x + 1 x r + 1 4.10 3 ) = +1 3 `,m ∗ ` `,m ` ` d d 2 αβ r Y M – 13 – ` F λF Z r Z = ) for a magnetostatic configuration. In this case, `,m = X ) and ( µναβ 1 `,m λ + 1 = d ( π ε 1 ˜ 4.10 ˜ 4 ` Q Q λ 2.24 µν J − Σ ) = d = x S ( I `,m ) can be related to the three form Noether current M ≡ Φ M λ 2.18 ˜ are the “magnetic multipole moments” given by Q ), we find `,m 4.10 M is the Levi-Civita symbol, and the gauge parameter ]) through the electic-magnetic duality ) in ( 30 multipole charges. These charges are blind to the magnetic field, and hence cannot µναβ 4.13 ε Let us finish this section by noting that since Maxwell theory is linear, one can su- The author is grateful to Jarah Evslin, Temple He, and Shahin Sheikh-Jabbari for useful discussions 6 perpose arbitrary magnetostatic andsolution. electrostatic solutions The to multipole obtain chargesbutions a ( respectively, general and stationary are blind to the other. Weon will this use section. these results in section Using ( The coefficients We observe that this istion. similar to Specifically, the the electric monopole resultwe charge ( compute corresponding to the magneticoutside charge the source, ( the magnetic field can be written as In the last equation, we have usedbe the written Bianchi identity as volume integrals two form, the charges ( which is conserved on-shell, whilecurrent the above charge corresponds to the where solutions of Laplace equation of the form We showed in previouselectric section thatuniquely the fix nontrivial the gauge field.of symmetries charges In are that order correspond associated to(see to to also overcome magnetic [ this multipoles. deficiency, we need to define a new set 4.2 Magnetostatics JHEP06(2017)080 ) (5.5) (5.6) (5.3) (5.4) (5.1) (5.2) 2.21 . Integrating 3 . to arrive at λ λ direction. Its current ≡ F z ) . λ ) × ∇ ) = 0 j λ B ρ, along the + , v × ∇ = ( j 2 , / µ λ 3 B ( j )) λ . · t 2 + ( ) 0 j n · ∇ da r · λ j E ( v − I γ , = ), using γ r − ( ) ∇ · E – 14 – h 3 ( λ = 2.21 µ × 1 + ( ) + q δ J ) v λ and use the time independence of s µ ( λ = ∂ = = λ Q ρ · ∇ E B + E ) ) by h + ( λ is not conserved individually, since the multipole charge can Q 2.2 ) ρ λ h ( ( λ d dt hypersurface and using the definition of hard and soft pieces of d Q dt and = µ is the position of the point charge. The particle produces electric and λ dt , representing only the hard piece of multipole current whose integral dx µ ˆ Q z const ρ d λj dt v t = = ≡ µ t ), we obtain j ) ) = h . Then multiply ( t ( λ ) ( is a combination of nontrivial residual symmetries found in section symmetries in more detail. µ h 2.22 0 ( λ J r λ Q `,m Consider a particle moving with a constant velocity λ magnetic fields is given by where A simple intuition that mayis stop that a one point to charge think withtime. of constant Before velocity multipole studying has moments the an as real increasingis dynamical conserved dipole trivial situations, charges moment let given growing us with the discuss facttransformation, this that it example. it will While can be this be illuminating in reverted to some aspects. the electrostatic case through a Lorentz This gives the transfer rate ofand multipole makes charge clear from the why electromagnetic the field hard to charge the source, 5.2 is not conserved. A preliminary example statement of themultipole conservation moment of i.e. multipolefreely charge. interpolate between Note electromagnetic field thatdefine and the charged matter. usual To expression seegives this for explicitly, This implies that theand time EM rate field of equals change minus of the multipole flux charge stored of in multipole both current the at source the boundary. This is the where this over a charge ( Before studying the charges infor electrodynamics, let us discuss the conservation5.1 law ( Conservation Let us expand the continuity equation ( 5 Electrodynamics JHEP06(2017)080 ) 5.6 (5.9) (5.7) (5.8) t , the total incoming R/v . ), the rate of transfer of = v R t 5.3 , > | v R t . The difference is nothing but qv , | < = qR | t z | , produced by a charged particle moving 2 v R xJ R ) 3 3 t d ( R is a unit radial vector from the charge’s z ). Therefore the change in dipole moment q Z q R n 1 3 3 1 5.7 qvt , - 1,0 = − Q 3 as in the previous section due to the soft part 0 = q R , – 15 – / and 2 3 1 0 , λ 2 ) = 1 / t q 1 ( · ∇ ) 0 − , s ) ( 1 j 2 ) = . There is no flux of charged particles at the boundary t x Q . v ( as expected. After a period ∆ 3 R ) 1 0 direction. The discontinuities occur when the particle enters and d , − h ( 1 = z qv ) = t Z 2 3 Q r ( 0 , when the particle escapes the integration surface, there is a − , 1 = (1 = Q γ R/v 0 , , 1 3 | ) = F 0 0 r r t − . The flux of dipole charge in this period, can be obtained using ( − r r | ( π q 4 . At = < R/v . The dipole charge within a sphere of radius qR | 2 3 t E ), leading to | 5.2 charge within the integration surface decays as These are summarized in figure when in ( flux is sudden outgoing flux ofthe hard soft charge charge by remaining the in the amount integration + surface. As the particle gets farther, the soft which is exactly the timeof the rate point of charge change is ofThe due total to ( the dipole absorption charge of isstored dipole off in charge by the from a field. the factor electromagnetic 2 Thereforeif field. the there total is charge a is fluxsurface not of constant is dipole in a charge time. at large This the sphere is boundary. at possible To only see this, assume that the integration which is linearly growingdipole in charge time. between the Meanwhile, source according and the to field ( is present position todipole the charge) observation is point. Now the dipole moment (the hard piece of Figure 1 with constant velocity along the exits the integration surface. where JHEP06(2017)080 ). R and 5.2 is a − (5.10) of the , there 1 and the A ∗ . Further `,m t of constant of radius Y ` B r > T S | = t . The important | is given by ( `,m S , as it is clear from λ F → ∞ fall in the region R . S as → ∞ B at a constant time slice , R and the flux λ R , both F + B cT . The reason is that ,S Σ − − Z S Q can be any given number). We first prove = R 0 ) = t − ( ( λ + – 16 – 0 Q Q − < t reside in region I as | t (+) λ | . Now let us compute the charge at two different times S Q is given by an integral over its boundary → ∞ T R = is taken to infinity i.e. , that is before and after the dynamics of the source. As mentioned t 0 ) computed at a sphere of radius R t, R . These two surfaces are connected by a timelike hypersurface Σ ( T > t Σ `,m ∂ is the timelike boundary between Q = . A distribution of charged matter (colored), which is stationary in far past and far future B where . Note that we have assumed that the source of radiation has compact support S T 2 As we mentioned before, associated with any residual symmetry before and after the radiation phase, and are given by surface integrals over = + where Σ However, the flux isfigure zero since no radiation can reach Σ as before, we assumet that before, the charge at point is that since spacetime. Using this we can show that that the charge isinfinite conserved number in of time, constraintsschematic and over picture then of the the show radiation problem. how produced this by conservation the lawsis source. impose a charge Figure generated by a dynamicaldynamics charged takes system. place Throughout insince this a an section, region eternal dynamical we of radiating assume spacetime systemwe that requires with consider an the compact infinite the source support. matter ofdynamics energy. This configuration happens Therefore is which in reasonable is the interval stationary in the region where radius (not drawn). Note that both 5.3 Infinite constraintsIn over radiation this section, we study how the multipole charge conservation constrains the radiation Figure 2 (regions I,III) and radiatingΣ in the region II. The charges are computed at two time slices Σ JHEP06(2017)080 `,m ˜ Q (5.15) (5.16) (5.12) (5.14) (5.11) (5.13) . Therefore 2 . λ J III + . Σ . The constant time hypersur- . Z + as shown in figure + . (+) `,m carried by the radiation is λ (+) `,m III + , J q M (+) `,m ) Σ II + rad `,m q , − − − . Σ ( `,m II + ) ) Q ) q Z − − Σ − + 1 ), i.e. ( ( `,m ( `,m `,m + 1 , + q ` ` I + Q + 1 λ M 2 + 1 4.5 ` J ` , the first term on the right hand side is zero, = 2 = I + – 17 – + 1 Σ + 1 = + 1 (+) `,m Z ` + 1 ` `,m ) 2 ` Q 4.1.2 ` J − = 2 ]. However, finding a precise relation is beyond the ( `,m = λ III + Q = ) J Σ 43 , ) Z + is given by ( rad Σ ( `,m rad 25 T Z ( `,m , Q ˜ − Q = 23 = through a volume integral over Σ t (+) λ , to obtain another set of constraints over the radiation Q (+) Q 4.2 denotes the multipole moments in the stationary phase before the dynamics. , we have found infinitely many constraints over the radiation produced during ) ) − naturally divides into three regions Σ ( ( `,m `,m q + ,M ) ( `,m In this paper,Maxwell we theory. studied We the showed thatLorenz conservation among laws gauge, the associated residual only gauge withcorrespond those transformations to residual solving surviving nontrivial the symmetries the conserved of proportional Laplace charges. to equation Interestingly, the and these multipole charges are moments turned of growing out the in to charged large be matter radius distribution, hence dubbed resembles the recent developments relating theberg asymptotic soft symmetries photon of QED theoremscope with [ of Wein- this paper. 6 Discussion Given merely the initialq and final stationarythe configuration dynamical of phase the of matter, the determined system, by without solving the equations of motion. This result Similarly, the same argumentdiscussed can in section be repeated for the magnetic multipole charges Therefore we find that the total multipole charge According to the discussionwhile in the section last term is where Now let us compute face Σ presence of radiation Moreover, the charge at in space and time. Accordingly, we obtain the conservation of multipole charges in the JHEP06(2017)080 ] to 23 ], we still expect that there is a 27 , 25 , 23 , 22 – 18 – ] that in static gauge, the multipole symmetries generate ) and their results. The reason is that there is a one to one 37 5.15 ], but to our knowledge never studied in relation with conservation 45 ]. 34 ] that the absorption rates of long wavelength radiation is determined by the 44 ]. It will be shown in [ 7 As we mentioned, multipole charges do arise in other non-covariant gauge condi- While there is qualitative difference in radial dependence of our residual symmetries The same study may be repeated in gravity where the gravitational multipole expansion Although the analysis in this paper was done for flat spacetime in four dimensions, Using the conservation of multipole charges, we found infinite number of constraints This analysis can be followed in different directions that we mention in the follow- Acknowledgments The author would like tocourse thank of specially this M.M. project. Sheikh-Jabbari Also forGiribet, I many T. am discussions He, grateful in M. to the Mirbabayi,paper. H. M. Afshar, I Pate S. and also Avery, G. D. appreciate Compère,J. Van Evslin, the den G. organizers Bleeken of for the their workshop comments on on quantum the aspects of black holes Interestingly, the conserved momenta of these geodesicsAn coincide with investigation the of multipole charges. thegeneral problem argument in on other the gaugebe equivalence found fixing of in conditions physical [ would results be in appealing. different gauge A conditions can the bulk may leadrelate to asymptotic a symmetries deeper of understanding future of and the past “antipodal null matching” infinity. tions used [ in [ a “vacuum moduli space”, whose geodesics represent physical electrostatic configurations. with the asymptotic symmetries considered inclose [ link between equation ( correspondence between the smooth solutionsan to arbitrary the function Laplace on equation the inside sphere. a sphere Also and the extension of asymptotic symmetries inside conservation of energy and large gauge symmetry charges. is well established [ laws of residual symmetries.cially The interesting constraints due over to the the recent gravitational detection radiation of can gravitational waves be from black espe- hole mergers. we expect that similararbitrary analysis dimensions. can be Innote carried case that out of a for asymptoticallyshown part asymptotically flat in of flat [ black radiation spacetimes holes can in geometries, be one absorbed should by the black hole. In this case, it was over the radiation produced byof the the charged source matter, which without can knowing be about in the general dynamics complicated. ing. While the electricsymmetries, multipole a charges first are principle Noether derivation charges of derived magnetic from charges residual remains gauge as an open issue. since the electromagnetic fieldis gives a exactly soft what contributionobviously makes to not the the conserved. multipole multipole The charge,in only charge and exception the conserved, this is charged the matter. while electricmultipole the Using monopole charges proportional which the multipole is to electric-magnetic moment only magnetic duality, stored is multipole we moments. also defined the magnetic as “multipole charges”. The multipole charge is not equal to electric multipole moment, JHEP06(2017)080 ] (A.4) (A.5) (A.1) (A.2) (A.3) tangent A ] ψ i λ δ ]. ]. physics/0503066 [ SPIRE IN SPIRE ][ IN . ][ B . ] (1918) 235 ) (A.6) . ψ B 2 ψ ] 2 λ 2 δ ψ λ [ H 2 1918 A λ B ] . G. δ ∂ [ ψ, δ ψ B B 1 1 A 1 ] λ ] λ δ ψ [ ), we conclude that hep-th/0111246 ψ H F ∂ λ gr-qc/9403028 1 [ δ A A [ λ ψ, δ [ AB ∂ ( δ ∂ A.1 [ , we arrive at ω AB AB B AB Σ AB A Gott. Nachr. δ Z , – 19 – ) = Ω (2002) 3 defines a Poisson bracket between functions over = Ω = Ω ψ = = = Ω = Ω 2 λ (1994) 846 } ) is translated to } } } 2 AB H 2 2 CB λ λ A λ ψ, δ 2.7 Ω B 633 ∂ 1 F,G ,H ,H ,H D 50 1 { 1 AC ), which permits any use, distribution and reproduction in 1 λ λ ψ, δ λ ( Covariant theory of asymptotic symmetries, conservation laws H H H { ω { { Some properties of Noether charge and a proposal for dynamical Σ Z Nucl. Phys. Phys. Rev. , , CC-BY 4.0 This article is distributed under the terms of the Creative Commons Invariant variation problems ]. ) and the fact that Ω A.2 SPIRE IN [ and central charges black hole entropy G. Barnich and F. Brandt, V. Iyer and R.M. Wald, E. Noether, [2] [3] [1] Attribution License ( any medium, provided the original author(s) and source are credited. References Translating back to spacetime notation using ( Open Access. Using ( Fortunately the Poisson bracket ofform charges of can the be inverse. computed This without is knowing the because explicit The inverse of the symplecticthe phase form space Ω through The right hand side is theto symplectic the two phase form space. contracted with Also two vectors equation [ ( (BMN) and SarAmadan club of Iran for the partialA support. Algebra of chargesHere in we covariant phase briefly space discusscoordinate the system over algebra the of infinite charges dimensional phase in space, the we covariant can phase write space. 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